Lines of Symmetry
Semester 4B · Chapter 16 · Lesson 5 · MATH-4B-C16-L05
Before you begin: prerequisite check
Recall: Basic congruence. Attempt these questions independently first.
State two defining geometric properties of a rhombus.
Show answer and reasoning
Use side lengths, parallelism, and right angles—not appearance.
Must every rhombus have two pairs of parallel opposite sides?
Show answer and reasoning
Squares, rectangles, rhombuses, and parallelograms all have both pairs of opposite sides parallel.
Can a square be a parallelogram? Justify.
Show answer and reasoning
Both opposite side pairs of the named shape are parallel.
Key vocabulary
- perpendicular
- meeting at a right angle
- parallel
- same direction and no intersection in a plane
- quadrilateral
- four-sided polygon
- line of symmetry
- line reflecting one matching half onto the other
Understand the concept
A line of symmetry divides a figure into two halves that match after one side is reflected onto the other. You can imagine folding the figure across the line: every point must overlap its partner, not just the corners. A non-square rectangle has two symmetry lines through its center. A square has four. A scalene triangle has none. To test a candidate line, check several points and their equal perpendicular distances from the line.
Fully worked examples and immediate checks
Read each example, predict how to begin, then follow its numerical explanation. The check below it is a fresh problem.
Worked example 1
A paper cutout has the shape of a square. How many reflection-symmetry lines does it have?
Square figures have 4 exact fold axes.
Check 1: try it yourself
A tile design has the shape of a square. How many reflection-symmetry lines does it have?
Show answer and reasoning
Square figures have 4 exact fold axes.
Worked example 2
A pattern on a card has the shape of a scalene triangle. How many reflection-symmetry lines does it have?
Scalene triangle figures have 0 exact fold axes.
Check 2: try it yourself
A tile design has the shape of an isosceles non-equilateral triangle. How many reflection-symmetry lines does it have?
Show answer and reasoning
Isosceles non-equilateral triangle figures have 1 exact fold axes.
Worked example 3
A pattern on a card has the shape of a square. How many reflection-symmetry lines does it have?
Square figures have 4 exact fold axes.
Check 3: try it yourself
A classroom window has the shape of a circle. How many reflection-symmetry lines does it have?
Show answer and reasoning
Circle figures have infinitely many exact fold axes.
Guided practice
A pattern on a card has the shape of an isosceles non-equilateral triangle. How many reflection-symmetry lines does it have?
Show answer and reasoning
Isosceles non-equilateral triangle figures have 1 exact fold axes.
A chalkboard diagram has the shape of an isosceles non-equilateral triangle. How many reflection-symmetry lines does it have?
Show answer and reasoning
Isosceles non-equilateral triangle figures have 1 exact fold axes.
A paper cutout has the shape of a non-square rectangle. How many reflection-symmetry lines does it have?
Show answer and reasoning
Non-square rectangle figures have 2 exact fold axes.
For an equilateral triangle drawn on a grid, explain why a proposed fold is a symmetry only when corresponding points overlap. How many possible symmetry axes are there?
Show answer and reasoning
Test matching perpendicular distances across the fold; this shape has 3.
Would rotating an isosceles non-equilateral triangle change the number of reflection-symmetry lines?
Show answer and reasoning
Rotation does not change the count; the shape retains 1 fold axes.
Independent practice
A classroom window has the shape of a square. How many reflection-symmetry lines does it have?
Show answer and reasoning
Square figures have 4 exact fold axes.
A classroom window has the shape of a non-square rectangle. How many reflection-symmetry lines does it have?
Show answer and reasoning
Non-square rectangle figures have 2 exact fold axes.
A paper cutout has the shape of an isosceles non-equilateral triangle. How many reflection-symmetry lines does it have?
Show answer and reasoning
Isosceles non-equilateral triangle figures have 1 exact fold axes.
For a square drawn on a grid, explain why a proposed fold is a symmetry only when corresponding points overlap. How many possible symmetry axes are there?
Show answer and reasoning
Test matching perpendicular distances across the fold; this shape has 4.
A chalkboard diagram has the shape of a non-square rectangle. How many reflection-symmetry lines does it have?
Show answer and reasoning
Non-square rectangle figures have 2 exact fold axes.
A tile design has the shape of a circle. How many reflection-symmetry lines does it have?
Show answer and reasoning
Circle figures have infinitely many exact fold axes.
A tile design has the shape of an equilateral triangle. How many reflection-symmetry lines does it have?
Show answer and reasoning
Equilateral triangle figures have 3 exact fold axes.
A pattern on a card has the shape of a circle. How many reflection-symmetry lines does it have?
Show answer and reasoning
Circle figures have infinitely many exact fold axes.
Challenge and transfer
Use a diagram or verbal justification where appropriate. The goal is transfer rather than speed.
A paper cutout has the shape of a scalene triangle. How many reflection-symmetry lines does it have? Explain or sketch the mathematical relationship.
Show answer and reasoning
Scalene triangle figures have 0 exact fold axes. State which quantities or geometric properties justify the method.
A learner claims: “Every diagonal of any rectangle is a line of symmetry.” Identify what is wrong or incomplete and give a mathematically correct explanation.
Show answer and reasoning
False: a non-square rectangle has only the horizontal and vertical center lines, not diagonal symmetry. Explain the relevant quantities or properties, not merely the final answer.
Exit check
A classroom window has the shape of a scalene triangle. How many reflection-symmetry lines does it have?
Show answer and reasoning
Scalene triangle figures have 0 exact fold axes.
For a non-square rectangle drawn on a grid, explain why a proposed fold is a symmetry only when corresponding points overlap. How many possible symmetry axes are there?
Show answer and reasoning
Test matching perpendicular distances across the fold; this shape has 2.
A tile design has the shape of a scalene triangle. How many reflection-symmetry lines does it have?
Show answer and reasoning
Scalene triangle figures have 0 exact fold axes.
Mastery check: Solve a new instance without hints, explain why the method works, and return later to confirm retention.